On the minimal length of extremal rays for Fano 4-folds

TL;DR

This paper proves that for certain smooth Fano 4-folds with birational contractions, the minimal intersection number of the anti-canonical divisor with rational curves is achieved on an extremal ray, confirming a key expectation in Fano geometry.

Contribution

It establishes that in smooth Fano 4-folds with birational contractions, the pseudo-index is realized on an extremal rational curve, advancing understanding of their geometric structure.

Findings

The minimal intersection number is attained on an extremal ray.

The result applies specifically to smooth Fano 4-folds with birational contractions.

Supports the conjecture relating pseudo-index and extremal rays in Fano manifolds.

Abstract

The minimum of intersection numbers of the anti-canonical divisor with rational curves on a Fano manifold is called pseudo-index. It is expected that the intersection number of anti-canonical divisor attains to the minimum on an extremal ray, i.e. there exists an extremal rational curve whose intersection number with the anti-canonical divisor equals the pseudo-index. In this note, we prove this for smooth Fano 4-folds having birational contractions.